## Franklin's Notes

### Chart on an affine space

Given an affine space $X$, we may "choose coordinates" for $X$ by first choosing an origin and then isomorphically mapping the resulting vector space onto $\mathbb R^n$. This is accomplished using a composite function called a chart: where the map $X\to T_a X$ is given by "fixing" the origin at $a$ by sending $x\mapsto (a,x)$ in the tangent space at $a$; the map $T_a X\to T$ is given by taking differences $(a,x)\mapsto d_a(x)=d(a,x)$; and the map $A:T\to \mathbb R^n$ is a "choice of basis". For clarity, $C_a$ is not used to convert $X$ into a vector space - just to assign it a coordinate system that is useful for labeling points.